Question

Prove: A nonempty subset C⊆R is closed if and only if there is a continuous function...

Prove:

A nonempty subset C⊆R is closed if and only if there is a continuous function g:R→R such that C=g-1(0).

Homework Answers

Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
let A be a nonempty subset of R that is bounded below. Prove that inf A...
let A be a nonempty subset of R that is bounded below. Prove that inf A = -sup{-a: a in A}
Let A be open and nonempty and f : A → R. Prove that f is...
Let A be open and nonempty and f : A → R. Prove that f is continuous at a if and only if f is both upper and lower semicontinuous at a.
Prove that if f: X → Y is a continuous function and C ⊂ Y is...
Prove that if f: X → Y is a continuous function and C ⊂ Y is closed that the preimage of C, f^-1(C), is closed in X.
Prove or provide a counterexample If A is a nonempty countable set, then A is closed...
Prove or provide a counterexample If A is a nonempty countable set, then A is closed in T_H.
let F : R to R be a continuous function a) prove that the set {x...
let F : R to R be a continuous function a) prove that the set {x in R:, f(x)>4} is open b) prove the set {f(x), 1<x<=5} is connected c) give an example of a function F that {x in r, f(x)>4} is disconnected
a)Suppose U is a nonempty subset of the vector space V over field F. Prove that...
a)Suppose U is a nonempty subset of the vector space V over field F. Prove that U is a subspace if and only if cv + w ∈ U for any c ∈ F and any v, w ∈ U b)Give an example to show that the union of two subspaces of V is not necessarily a subspace.
if f: D - R be continuous, and D is close, then F(D) is closed. prove...
if f: D - R be continuous, and D is close, then F(D) is closed. prove or give counterexample
Problem 6. For a closed convex nonempty subset K of a Hilbert space H and x...
Problem 6. For a closed convex nonempty subset K of a Hilbert space H and x ∈ H, denote by P x ∈ K a unique closest point to x among points in K, i.e. P x ∈ K such that ||P x − x|| ≤ ||y − x||, for all y ∈ K. First show that such point P x exists and unique. Next prove that all x, y ∈ H ||P x − P y|| ≤ ||x −...
Let f, g : X −→ C denote continuous functions from the open subset X of...
Let f, g : X −→ C denote continuous functions from the open subset X of C. Use the properties of limits given in section 16 to verify the following: (a) The sum f+g is a continuous function. (b) The product fg is a continuous function. (c) The quotient f/g is a continuous function, provided g(z) != 0 holds for all z ∈ X.
We know that any continuous function f : [a, b] → R is uniformly continuous on...
We know that any continuous function f : [a, b] → R is uniformly continuous on the finite closed interval [a, b]. (i) What is the definition of f being uniformly continuous on its domain? (This definition is meaningful for functions f : J → R defined on any interval J ⊂ R.) (ii) Given a differentiable function f : R → R, prove that if the derivative f ′ is a bounded function on R, then f is uniformly...
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT